Problem 46 - College Algebra and Calculus: An Applied Approach

Question

Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. $$x^{4}-6 x^{2}-72$$

Step-by-Step Solution

Answer

The fully factorized form of the polynomial is \((x- \sqrt[4] {72})(x + \sqrt[4] {72})(x - \sqrt[4] {72}i)(x + \sqrt[4] {72}i)\).

Step 1: Express as irreducible factors over the rationals

Recognize the polynomial as a quadratic in the form of \(a^2 - b^2\). First substitute \(a = x^2\) and \(b = \sqrt {72}\). That renders us to \(a^2 - b^2 = (a - b)(a + b)\). Thus, our polynomial becomes \((x^2 - \sqrt {72})(x^2 + \sqrt {72})\).

Step 2: Express as linear and quadratic factors that are irreducible over the reals

Expression \(x^2 - \sqrt {72}\) can be decomposed into linear factors over the reals as \((x- \sqrt[4] {72})(x + \sqrt[4] {72})\) and \(x^2 + \sqrt {72}\) is already a quadratic factor that is irreducible over the reals.

Step 3: Factorize completely

The term \(x^2 + \sqrt {72}\) has complex solutions which are \(x = \pm\sqrt[4] {72}i\) (where \(i\) is the imaginary unit). Therefore, \(x^2 + \sqrt {72}\) can be factored into \((x - \sqrt[4] {72}i)(x + \sqrt[4] {72}i)\) in the complex domain.

Key Concepts

Irreducible factors

Understanding irreducible factors is crucial for polynomial factorization. These are factors that cannot be broken down any further within a given set of numbers. In this exercise, we look at irreducible factors over different number systems. For example, over the rationals, if you consider a polynomial like the given one, you apply the difference of squares which states:

• For any two numbers \(a\) and \(b\), \(a^2 - b^2 = (a-b)(a+b)\).

Thus, by recognizing the polynomial as \((x^2 - 72^{1/2})(x^2 + 72^{1/2})\), you have expressed it as irreducible factors over the rationals. Each factor here is not further factorable by any rational number. This is an important step because it shows how far you can decompose a polynomial under the constraint of rational numbers. It sets the foundation for understanding different levels of factorization based on the numerical set in which the polynomial is interpreted.

Rational numbers

Rational numbers are numbers that can be expressed as a fraction \(\frac{p}{q}\) with both \(p\) and \(q\) integers and \(q eq 0\). When factoring polynomials over the rationals, you try to express the polynomial as a product of polynomials with rational coefficients. For example, in our exercise, when we factor the polynomial \(x^4 - 6x^2 - 72\) over the rationals, we find that its irreducible factors are \( (x^2 - \sqrt{72})(x^2 + \sqrt{72}) \). Both are expressed in terms of the quadratic roots, and this demonstrates how understanding the capabilities of rational numbers helps in breaking down the polynomial. This form allows for examining polynomial properties without fully solving for roots, which might otherwise require moving outside the set of rationals, to real or complex solutions.

Real numbers

Real numbers include all the rational numbers plus the irrational numbers, which are numbers that cannot be expressed as simple fractions, like \(\pi\) or \( \sqrt{2} \). In polynomial factorization, using real numbers allows more flexibility than rationals because you can incorporate irrational numbers.The factorization of \(x^4 - 6x^2 - 72\) over the real numbers involves expressing parts of the polynomial as linear factors, meaning factors of degree one. In this exercise, you saw how \(x^2 - \sqrt{72}\) can be split into linear factors that include irrational numbers: \((x - \sqrt[4]{72})(x + \sqrt[4]{72})\). The polynomial \(x^2 + \sqrt{72}\) remains irreducible over the real numbers since its roots are not real. This shows how real numbers help further break down polynomials into factors that are easier to interpret regarding multiplication and are more practical within the real number system.

Complex numbers

Complex numbers extend the real numbers by introducing the imaginary unit \(i\), where \(i^2 = -1\). This expansion allows for solving every polynomial equation to its fullest extent, often revealing roots that aren't apparent when only using real numbers.In our task, the polynomial \(x^2 + \sqrt{72}\) is irreducible over the reals because its roots are complex. For complete factorization, you recognize that you can express every real and complex solution. Thus, it becomes \((x - \sqrt[4]{72}i)(x + \sqrt[4]{72}i)\). This complex factorization illustrates the power and necessity of complex numbers, especially in higher mathematics where real numbers alone cannot provide all solutions. Understanding complex numbers enhances our comprehension of polynomial behavior across different domains.